Bounding the Norm of a Log-Concave Vector Via Thin-Shell Estimates

Chaining techniques show that if X is an isotropic log-concave random vector in \(\mathbb{R}^{n}\) and Γ is a standard Gaussian vector then $$\displaystyle{\mathbb{E}\Vert X\Vert \leq Cn^{1/4}\mathbb{E}\Vert \varGamma \Vert }$$ for any norm \(\Vert \cdot \Vert\), where C is a universal constant. Using a completely different argument we establish a similar inequality relying on the thin-shell constant $$\displaystyle{\sigma _{n} =\sup {\Bigl ( \sqrt{\mathrm{Var }(\vert X\vert )};\ X\text{ isotropic and log-concave on }\mathbb{R}^{n}\Bigr )}.}$$ In particular, we show that if the thin-shell conjecture σ n = O(1) holds, then n 1∕4 can be replaced by log(n) in the inequality. As a consequence, we obtain certain bounds for the mean-width, the dual mean-width and the isotropic constant of an isotropic convex body. In particular, we give an alternative proof of the fact that a positive answer to the thin-shell conjecture implies a positive answer to the slicing problem, up to a logarithmic factor.